3.67 \(\int \frac {1}{(4+12 x+9 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=25 \[ -\frac {1}{6 (3 x+2) \sqrt {9 x^2+12 x+4}} \]

[Out]

-1/6/(2+3*x)/((2+3*x)^2)^(1/2)

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Rubi [A]  time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {607} \[ -\frac {1}{6 (3 x+2) \sqrt {9 x^2+12 x+4}} \]

Antiderivative was successfully verified.

[In]

Int[(4 + 12*x + 9*x^2)^(-3/2),x]

[Out]

-1/(6*(2 + 3*x)*Sqrt[4 + 12*x + 9*x^2])

Rule 607

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*(a + b*x + c*x^2)^(p + 1))/((2*p + 1)*(b + 2
*c*x)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (4+12 x+9 x^2\right )^{3/2}} \, dx &=-\frac {1}{6 (2+3 x) \sqrt {4+12 x+9 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 0.80 \[ -\frac {3 x+2}{6 \left ((3 x+2)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + 12*x + 9*x^2)^(-3/2),x]

[Out]

-1/6*(2 + 3*x)/((2 + 3*x)^2)^(3/2)

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fricas [A]  time = 0.96, size = 14, normalized size = 0.56 \[ -\frac {1}{6 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x+4)^(3/2),x, algorithm="fricas")

[Out]

-1/6/(9*x^2 + 12*x + 4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x+4)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.05, size = 17, normalized size = 0.68 \[ -\frac {3 x +2}{6 \left (\left (3 x +2\right )^{2}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*x^2+12*x+4)^(3/2),x)

[Out]

-1/6*(3*x+2)/((3*x+2)^2)^(3/2)

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maxima [A]  time = 2.89, size = 9, normalized size = 0.36 \[ -\frac {1}{6 \, {\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x+4)^(3/2),x, algorithm="maxima")

[Out]

-1/6/(3*x + 2)^2

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mupad [B]  time = 0.17, size = 21, normalized size = 0.84 \[ -\frac {\sqrt {9\,x^2+12\,x+4}}{6\,{\left (3\,x+2\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(12*x + 9*x^2 + 4)^(3/2),x)

[Out]

-(12*x + 9*x^2 + 4)^(1/2)/(6*(3*x + 2)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (9 x^{2} + 12 x + 4\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x**2+12*x+4)**(3/2),x)

[Out]

Integral((9*x**2 + 12*x + 4)**(-3/2), x)

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